G. Koru Yücekaya, H.H. Hacısalihoğlu
)
,
HOLDITCH’S THEOREM FOR CIRCLES IN 2-DIMENSIONAL EUCLIDEAN SPACE
Gülay KORU YUCEKAYA*, H. Hilmi HACISALİHOĞLU**
* Ahi Evran University, Faculty of Art and Sciences, Department of Mathematics, 40200, Kırşehir, TURKEY, gkoruyucekaya@yahoo.com.tr
**
Ankara University, Faculty of Sciences, Department of Mathematics, Tandoğan, 06100, Ankara, TURKEY, hacisali@science.ankara.edu.trGeliş Tarihi: 29.12.2008 Kabul Tarihi: 09.03.2009 ABSTRACT
The present study expresses and proves Holditch’s Theorem for two different circles in two-dimensional Euclidean space through a new method.
Key Words: Affine space, Euclidean space, Euclidean circle, Holditch’s Theorem
2-BOYUTLU ÖKLİD UZAYINDA ÇEMBER İÇİN HOLDITCH TEOREMİ
ÖZET
Bu çalışmada, 2-boyutlu Öklid uzayında farklı iki çember için Holditch Teoremi yeni bir metodla ifade ve ispat edilmiştir.
Anahtar Kelimeler: Afin Uzay, Öklid uzayı, Öklid çemberi, Holditch Teoremi
1. INTRODUCTION
In this section we give basic definitions and theorems used in this study.
Definition 1.1 Let A be a non-empty set and V be a vector space on a field F.
i. For ∀P Q R A, , ∈ , f P Q
(
,)
+ f Q R(
,)
= f P R(
.(
,)
f P Q =α α V
∀ ∈ , there is a unique point Q A∈
ii. For ∀ ∈P A and so that .
If there is a function f AxA: →V, satisfying above propositions, then A is called an Affine space associated to V [1].
Definition 1.2 Let A be a real affine space and let V be a vector space associated to A. Using Euclidean inner product operation on V,
(
1) (
11
, , ,..., , ,...,
n
i i n n
)
X Y x y X x x Y y y
→ → → →
=
∑
= =, :VxV→IR,
we can define the metric concepts such as distance and angle in A. Therefore, Affine space A is called a in Euclidean space and is denoted by A E= n [1].
:IRn →IR+ Definition 1.3 The transformation defined by
,
X X X
→ → →
=
X→ is called the norm of the vector .
X→
Definition 1.4 For ∀X Y→ →, ∈IRn , the measure of the angle between and is the real number Y→ θ derived from
, cos
X Y X Y θ
→ →
→ →
= .
, ,
AB→ =c AC→ =b BC→ a ABCΔ
Definition 1.5 (The Pythagoras Theorem) In a right triangle called , if =
2
, then a2 =b2+c (Figure 1.1) [2].
A
Figure 1.1 Right triangle ABCΔ
C c
a
b
B
Definition 1.6 The set of the points in which are at a distance r from a point M is called a circle with center M and radial length r Euclid circle and is denoted by
E2
: , consta
C=⎧⎨X MX→ =r r= ⎫⎬
⎩ nt [2]. ⎭
Definition 1.7 The area of a circle with radius r and with point A on it is ( A) 2
A C =πr [2].
Theorem 1.1 (Holditch’s Theorem) Let a chord with constant lengths of a b+ of a closed convex curve α be divided by a point on it into two segments with and b as lengths. Let us move the end-points of the chord so that they will entirely trace the curve. Then, the difference between the sizes of the area bounded by the closed curve drawn by point P and that bounded by the main convex curve
a P
πab is [3].
α
G. Koru Yücekaya, H.H. Hacısalihoğlu 2. THE CLASSICAL HOLDITCH THEOREM
Theorem 2.1 Let an AB chord with a constant length of a b+ on a circle (C) with a radius r in Euclidean plane be divided by a point D into two segments with lengths of a and b, respectively. When the end-points A and B of the chord draw the circle in full, then geometric location of D forms an inner circle (Figure 2.1).
E2
(C)
A
M
O
B a
.
b DFigure 2.1 Geometric location of D on the chord 2 constant
AB = + =a b l=
Proof. . Let M be the midpoint of AB. Then,
2 . AD a
BD b
MA MB a b l
=
=
= = + =
From the right triangle OMΔA
2 2
OA = OM + MA2
or
2 2
OM = OA − MA2
2 2
2 r ⎛a b+ ⎞
= − ⎜ ⎟
⎝ ⎠
MD = − b l
2 b a b+
= −
2 b a− = .
Similarly, from the right triangle OMDΔ
2 2
OD = OM + MD2
2 2
2
2 2
a b b a
r ⎛ + ⎞ ⎛ −
= −⎜⎝ ⎟⎠ +⎜⎝
⎞⎟
⎠ r ab
= −
2 .
Since a, b and r are constant, OD is also constant. Therefore, the geometric location of D is a circle with a centre O and a radial length of r2−ab.
Theorem 2.2 Let a chord AB with a constant length of a b+ on a circle (C) with a radius R in Euclidean plane be divided by a point D into two segments with lengths of a and b, respectively. When the end-points A and B of the chord draw the circle in full, the size of the ring-shaped region between the orbit of D (inner circle) and circle (C) is independent from the radial length of circle (C).
E2
z= r2−ab
Proof. For the radial length of the inner circle is and its two chords AB and EF intersecting at point D of circle (C), let the chord EF be the diameter of (C). The chords AB and EF are divided by D into line segments whose lengths are a, b and r+z, r-z, respectively (Figure 2.2).
A
B F
E r
z
r-z
r+z
r
(C) O
D
a
b
Figure 2.2 Two chords intersecting in circle (C)
Since the triangles BDEΔ and FDAΔ in the interior of circle (C) are similar triangles, BD DE
FD = DA b r r z a
= +
−
z
2 2
ab r= −z
Since the area of circle (C) is A C( )=πr2 and the area of the inner circle is πz2, the area of the ring-shaped region between these two circles is found as
( )
2 2 2 2
r z r z
π −π =π −
πab
=
G. Koru Yücekaya, H.H. Hacısalihoğlu
Therefore, the size of the ring-shaped region that falls between the orbit of D and circle (C) is independent from the radial length of circle (C).
Corollary 2.1 The size of the ring-shaped region that falls between the orbit of D and circle (C) is dependent on the selection of point D on the chord; that is, on the segments with lengths of a and b.
3. HOLDITCH’S THEOREM FOR TWO DIFFERENT CIRCLES IN A 2-DIMENSIONAL EUCLIDEAN SPACE
Theorem 3.1 For a circle C with a radial length of R a b+ + and a circle C′ with radius , let an AB rod with a constant length of with end B attached to circle C, and the other end A attached to circle by a joint, be divided by point X on it into two segments with lengths of a and b, respectively. When the rod with the constant length of draws the circles C and C
( )
R< R a b+ + constant ,
a b+ = C′
constant ′
a b+ = with its end-points within these
circles C and C , the geometric location of X forms another inner circle (Figure 3.1). During this motion, the relation between the regions bounded by circles is independent from the selection of circles C and C .
′
′
A X
B
O a b
R
C′
C
.
.Figure 3.1 Circles in E2 Proof.
AX→ =a, XB→ =b, OX→ = +R a
The area of circle C with a radial length R and with a point A on it is ′ ( A) 2
A C =πR
The area of circle C with a radial length R a b+ + and with a point B on it is
( )
2( B)
A C =π R a+ + b
Therefore, the area of the circle, which is the geometric location of point X is
( )
X( )
2A C =π R+ a
)
Thus,
( ) (
2 2( B) ( X)
A C −A C =π⎡⎣ R a b+ + − R a+ ⎤⎦ =πb⎡⎣2
(
R a+)
+b⎤⎦( )
2( A) ( X) 2
A C −A C =π⎡⎣R − R a+ ⎤⎦ = −πa R a
(
2 +)
and by adding these two equations side by side, we get
( )
2 2( )
2( A) ( B) 2 ( X) 2
A C +A C − A C =π⎡⎣ R a b b+ + +R − R a+ ⎤⎦
2 2 2 2
2Rb 2ab b R R 2aR a
π⎡ ⎤
= ⎣ + + + − − − ⎦
( )
2 22R b a 2ab b a
π⎡ ⎤
= ⎣ − + + − ⎦
( )( )
2ab b a b a 2R π
= ⎡⎣ + − + + ⎤⎦
( ) ( )
( A) ( B) 2 ( X) 2 2 A CA
A C A C A C π ab b a b a
π
⎡ ⎛ ⎞⎤
⎢ ⎜ ⎟⎥
+ − = + − + +
⎜ ⎟
⎢ ⎝ ⎠⎥
⎣ ⎦
Corollary 3.1 The relation between the regions bounded by circles with radial lengths of and is a and b; in other words, it is independent from the rod’s motion on the circles.
R a b+ + ,
R R a+
REFERENCES
[1] Hacısalihoğlu, H. H., “Diferential Geometry 1st. Ed.”, İnönü University Faculty of Arts and Science Publications, Mat. No.2, Malatya,Turkey, p 895 (1983).(Turkish)
[2] Hacısalihoğlu, H. H., “Analytic Geometry in 2 and 3 dimensional Spaces 2nd. ed.”, Gazi University Faculty of Arts and Science Publications, No.6, Ankara, p 528, (1984).(Turkish)
[3] Holditch, H., “Geometrical Theorem”, The Quarterly Journal Of Pure And Applied Math., Vol. II, p. 38, London, (1858).